Difference between revisions of "Store:FLen04"

Latest revision as of 14:53, 19 October 2022

Set operators

Given the whole universe $U$ we indicate with $x$ its generic element so that $x\in U$ ; then, we consider two subsets $A$ and $B$ internal to $U$ so that $A\subset U$ and $B\subset U$

	Union: represented by the symbol $\cup$ , indicates the union of the two sets $A$ and $B$ $(A\cup B)$ . It is defined by all the elements that belong to $A$ and $B$ or both: $(A\cup B)=\{\forall x\in U\mid x\in A\land x\in B\}$
	Intersection: represented by the symbol $\cap$ , indicates the elements belonging to both sets: $(A\cap B)=\{\forall x\in U\mid x\in A\lor x\in B\}$
	Difference: represented by the symbol $-$ , for example $A-B$ shows all elements of $A$ except those shared with $B$
	Complementary: represented by a bar above the name of the collection, it indicates by ${\bar {A}}$ the complementary of $A$ , that is, the set of elements that belong to the whole universe except those of $A$ , in formulas: ${\bar {A}}=U-A$

The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid. Remember that in classical logic, given the set $A$ and its complementary ${\bar {A}}$ , the principle of non-contradiction states that if an element belongs to the whole $A$ it cannot at the same time also belong to its complementary ${\bar {A}}$ ; according to the principle of the excluded third, however, the union of a whole $A$ and its complementary ${\bar {A}}$ constitutes the complete universe $U$ .

In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.

@@ Line 1: / Line 1: @@
-===<!--70-->Set operators===
+===Set operators===
-<!--71-->Given the whole universe <math>U</math> <!--72-->we indicate with <math>x</math> <!--73-->its generic element so that <math>x \in U</math>; <!--74-->then, we consider two subsets <math>A</math> and <math>B</math> <!--75-->internal to <math>U</math> <!--76-->so that <math>A \subset U</math> <!--77-->and <math>B \subset U</math>
+Given the whole universe <math>U</math> we indicate with <math>x</math> its generic element so that <math>x \in U</math>; then, we consider two subsets <math>A</math> and <math>B</math> internal to <math>U</math> so that <math>A \subset U</math> and <math>B \subset U</math>
 {|
 |[[File:Venn0111.svg|left|80px]]
-|'''<!--78-->Union:''' <!--79-->represented by the symbol <math>\cup</math>, <!--80-->indicates the union of the two sets <math>A</math> <!--81-->and <math>B</math> <math>(A\cup B)</math>. <!--82-->It is defined by all the elements that belong to <math>A</math> <!--83-->and <math>B</math> <!--84-->or both:
+|'''Union:''' represented by the symbol <math>\cup</math>, indicates the union of the two sets <math>A</math> and <math>B</math> <math>(A\cup B)</math>. It is defined by all the elements that belong to <math>A</math> and <math>B</math> or both:
 <math>(A\cup B)=\{\forall x\in U \mid x\in A \land x\in B\}</math>
 |-
 |[[File:Venn0001.svg|sinistra|80px]]
-|'''<!--85-->Intersection:''' <!--86-->represented by the symbol <math>\cap</math>, <!--87-->indicates the elements belonging to both sets:
+|'''Intersection:''' represented by the symbol <math>\cap</math>, indicates the elements belonging to both sets:
 <math>(A\cap B)=\{\forall x\in U \mid x\in A \lor x\in B\}</math>
 |-
 |[[File:Venn0010.svg|left|80px]]
-|'''<!--88-->Difference:''' <!--89-->represented by the symbol <math>-</math>, <!--90-->for example <math>A-B</math> <!--91-->shows all elements of <math>A</math> <!--92-->except those shared with <math>B</math>
+|'''Difference:''' represented by the symbol <math>-</math>, for example <math>A-B</math> shows all elements of <math>A</math> except those shared with <math>B</math>
 |-
 |[[File:Venn1000.svg|left|80px]]
-|'''<!--93-->Complementary:''' <!--94-->represented by a bar above the name of the collection, it indicates by <math>\bar{A}</math> <!--95-->the complementary of <math>A</math>, <!--96-->that is, <!--97-->the set of elements that belong to the whole universe except those of <math>A</math>, <!--98-->in formulas: <math>\bar{A}=U-A</math><br />
+|'''Complementary:''' represented by a bar above the name of the collection, it indicates by <math>\bar{A}</math> the complementary of <math>A</math>, that is, the set of elements that belong to the whole universe except those of <math>A</math>, in formulas: <math>\bar{A}=U-A</math><br />
 |}
-<!--99-->The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid. <!--100-->Remember that in classical logic, given the set <math>A</math> <!--101-->and its complementary <math>\bar{A}</math>, <!--102-->the principle of non-contradiction states that if an element belongs to the whole <math>A</math> <!--103-->it cannot at the same time also belong to its complementary <math>\bar{A}</math>; <!--104-->according to the principle of the excluded third, however, the union of a whole <math>A</math> <!--105-->and its complementary <math>\bar{A}</math> <!--106-->constitutes the complete universe <math>U</math>.
+The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid. Remember that in classical logic, given the set <math>A</math> and its complementary <math>\bar{A}</math>, the principle of non-contradiction states that if an element belongs to the whole <math>A</math> it cannot at the same time also belong to its complementary <math>\bar{A}</math>; according to the principle of the excluded third, however, the union of a whole <math>A</math> and its complementary <math>\bar{A}</math> constitutes the complete universe <math>U</math>.
-<!--107-->In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.
+In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.